Five notes on asymptotic prime divisors (Q1070286)

Let I be an ideal in a Noetherian ring R, \(A^*(I)=Ass R/I^ n\), and \(\bar A^*(I)=Ass R/(I^ n)_ a\) (the integral closure of \(I^ n)\), both definitions for all large n. \(\bar A^*(I)\subseteq U(I)\subseteq A^*(I)\), where U(I) is a set which is in many ways analogous to \(\bar A^*(I)\) (but whose definition is a bit complicated). These notes are related to various known results concerning these sets. The first is an application to going-down. The second concerns characterizing when a locally complete intersection ideal is a radical ideal. The third shows that three theorems known to hold for \(\bar A^*(I)\) have analogues for U(I), one of them being if I and J are regular ideals, and \(P\in U(I)\), then \(P\in U(J)\). The fourth note shows that for certain filtrations \((I_ n| \quad n\geq 1)\) related to the filtration \((I^ n| \quad n\geq 1)\) of powers of I, the sequence Ass R/I\({}_ n\) is monotonically increasing (unlike Ass R/I\({}^ n\) which is not). The final note concerns a regular principal ideal bR, and primes in Ass R/bR- U(bR).